

Unfortunately, problem (REF ) is very challenging to handle due to the following reasons: i) the objective function (REF ) is max-min function and non-differentiable which hamper the algorithm development, ii) the fractional terms in objective function (REF ) are non-convex, iii) the unit modulus constraint for the RIS phase-shifts in () is also non-convex. In order to tackle these difficulties, the max-min function is equivalently transformed into min-max form to apply the log-sum-exp approach and Riemannian-manifold algorithm. Then, we reformulate the fractional terms in (REF ) into a form with decoupled numerator and denominator by applying Dinkelbach's transform [1]}. Therefore, by introducing a new auxiliary variable \(\varpi _{\mathrm {r}}\) , the transformed problem can be expressed as
\(\min _{\mathbf {\phi }_{\mathrm {r}},\varpi _{\mathrm {r}}}&\max _{k\in \mathcal {K}_{\mathrm {r}}}~\left\lbrace f_{k}-\varpi _{\mathrm {r}} g_{k}\right\rbrace  \\\mathrm {s.t.}& ~~|\mathbf {\phi }_{\mathrm {r},m}|=1, ~~\forall m,\) 
